Giulia Faggio, Kjell G. Salvanes & John Van Reenen turn to heterogeneity in firm productivity to explain wage inequality:

[M]uch of the increase in individual wage inequality in the UK occurred between firms within the same industry (between-firm component) instead of within firms (within-firm component). This is an important finding when looking for ‘culprits’ of wage inequality. It says that little of extra inequality has come from a change in the way firms treat their own workers. The main source is the change in firm-level productivity. This implies that understanding the evolution of productivity distribution between firms may be critical in understanding the evolution of wage distribution (we also show that the correlation between wages and productivity has become more important over time)…

In terms of policy, this suggests that the causes of rising inequality are primarily structural and related to new technology rather than to trade or institutions. Thus greater trade protectionism or the re-energising of unions may do relatively little to reverse the increase in inequality.

But is trade orthogonal to the distribution of firm productivity? If one thinks of the Melitz (2003) model, then a decrease in trade costs eliminates the least productive firms and truncates the range of support for firms that survive. However, due to the fractal-like property of the Pareto distribution, the truncated distribution will still have same skewness as the initial distribution of productivity levels. For at least some measures of inequality, therefore, the distribution of productivity is independent of trade policy, even if trade costs determine the aggregate level of productivity through selection effects.

But that’s just a one-minute sketch using the most popular model of firm heterogeneity (and firms all pay the same wage in Melitz (2003)!). If the distribution of productivity is critical to wage inequality, then economists have renewed reason to investigate the relationship between trade and firm-level productivity.